7350
J. Phys. Chem. B 1997, 101, 7350-7359
Three-Pulse Photon Echo Measurements on LH1 and LH2 Complexes of Rhodobacter sphaeroides: A Nonlinear Spectroscopic Probe of Energy Transfer Ralph Jimenez,† Frank van Mourik,‡ Jae Young Yu,† and Graham R. Fleming*,† Department of Chemistry and the James Franck Institute, The UniVersity of Chicago, 5735 South Ellis AVenue, Chicago, Illinois 60637, and Department of Physics and Astronomy, The Free UniVersity of Amsterdam, de Boelelaan 1081, 1081 HV Amsterdam, The Netherlands ReceiVed: January 23, 1997; In Final Form: April 8, 1997X
Three-pulse echo peak shift measurements were performed on the B875 and B850 bands of detergent-isolated LH1 and LH2 complexes at room temperature. The peak shifts are much larger and decay much faster than typically observed for dye molecules in solution. Simulations of the peak shifts based on the optical transition frequency correlation function, M(t), are presented. M(t) includes contributions from rapid protein fluctuations, vibrational motion, and energy transfer. The model reproduces the room temperature absorption spectra of B850 and B875, shows that the coupling of electronic and nuclear degrees of freedom is much weaker than for dyes in solution, and identifies contributions to the line shapes that may be important to the energy transfer processes. The implications of these results for the extent of electronic delocalization in LH1 and LH2 are also discussed. Although the role of coherence transfer still needs to be understood, the results are shown to be consistent with the use of weak-coupling excitation transfer models of B850 and B875.
I. Introduction In photosynthesis, most of the light energy is absorbed by antenna complexes and then transferred to the reaction center with very high overall efficiency. The absorbed energy migrates over many pigments as an electronic excitation before being trapped by the reaction center in a process whose overall time scale is ∼50 ps.1 The individual excitation transfer steps, therefore, occur on subpicosecond time scales. In photosynthetic purple bacteria, there are two types of lightharvesting antenna, core (LH1) and peripheral (LH2) lightharvesting complexes. The chromophores in LH1 and LH2 are bacteriochlorophylls (Bchl) and carotenoids. Here, we will focus on the dynamics of the Bchl pigments in bacteriochlorophyll a-containing bacteria. The LH1 complexes contain only one type of Bchl absorbing at ∼875 nm (B875), whereas LH2 complexes contain two types of Bchl, absorbing at 800 nm (B800) and 850 nm (B850). Recently, high-resolution crystal structures of some LH2 complexes have become available. The R-helices of the protein coordinate the Bchl into ringlike structures with very small distances between pigments.2 For example, the crystal structure of LH2 from Rhodopseudomonas acidophila shows a basic unit with 9-fold symmetry.3 The Rβpolypeptides coordinate 18 B850 pigments into a closely interacting ring. The spacing between the chlorin rings is slightly staggered, alternating between 3.5 and 3.9 Å.4 The B800 pigments are separated by ∼21 Å, and the closest approach between B800 and B850 is 17.6 Å. The crystal structure of LH2 from Rhodospirillum molischianum shows similar features in a ring with 8-fold symmetry.5 Twodimensional electron diffraction of LH1 reveals a larger ring very similar to B850 with 32 Bchl, possibly segregated into pairs.6 However, the resolution of this study is not sufficient to specify precisely the distances between pigments. * To whom correspondence should be addressed. Permanent address: Department of Chemistry, University of California, Berkeley, Berkeley, CA 94720. † The University of Chicago. ‡ The Free University of Amsterdam. X Abstract published in AdVance ACS Abstracts, August 15, 1997.
S1089-5647(97)00299-X CCC: $14.00
1. Related Work. Many time-resolved studies have focused on determining the time scale of excitation transfer within these complexes at room temperature. Pump-probe techniques have been used to study B800 to B800 and B800 to B850 excitation transfer.7-9 In LH2, excitation transfer from B800 to B850 takes 600-700 fs at room temperature.7,8,13 Transfer within the B800 group of cofactors takes approximately 700 fs.9 Since the distances separating the pigments are large compared to the dimensions of the Bchl molecules, these processes are probably well described with weak-coupling theories of energy transfer such as Fo¨rster theory.10 The mode of energy transfer within the B850 and B875 pigments is less clear, because the electronic structure of these aggregates is not yet well understood. One line of thought is suggested by the finding that the LH1 antenna can be reversibly dissociated into a subunit that has the properties of a strongly interacting dimer of Bchl.11 This result has prompted the hypothesis that excitation transfer within B850 and B875 can be described as hopping from dimer to dimer.12,13,23 The extent to which this picture is accurate, or whether larger degrees of delocalized excitation (excitons) must be considered, are topics of current debate. For example, Sauer and co-workers have performed calculations of the optical absorption and circular dichroism (CD) spectra of LH2, using the crystallographic coordinates of Rps. acidophila.20 From simulations of the CD spectra, they suggest that the B850 electronic state is delocalized over nearly the entire ring at room temperature. Kennis et al. have measured subpicosecond absorbance difference spectra of LH2 complexes from Rhodobacter sphaeroides and Chromatium tepidum in which they interpret the unusually large absorbance changes in B850 (which are a factor of 4 larger than in B800) to be the result of excitation delocalized over the whole ring.17 In another experimental study, Pullerits et al. have modeled their absorbance difference spectra of B850 from Rb. sphaeroides with an exciton calculation including two excited electronic states of each Bchl.18,19 By fitting the shape of the spectral features, they conclude the excitation to be delocalized over 4 ( 2 Bchl. Monshouwer et al. measured the radiative rate of LH1 and LH2 and found values that are a factor of 2-3 higher than monomeric Bchl.21 None of these studies modeled © 1997 American Chemical Society
LH1 and LH2 Complexes of Rhodobacter sphaeroides the combined influence of vibrations and inhomogeneous broadening on the delocalization length. An attempt to consider all these effects was made in a theoretical study in which the extent of delocalization was calculated by a Green’s function approach as a function of electronic coupling, homogeneous line width, and inhomogeneous width.22 Recently, fluorescence depolarization measurements have been performed on the LH1 and LH2 complexes of Rb. sphaeroides.12,13 These studies interpreted the time scale of depolarization in terms of site-to-site hopping within the ringlike structures suggested by crystallography, thus revealing the time scale of the hopping process. For both B875 and B850, multiexponential depolarizations were observed. For example, in LH1 time constants of approximately 100 and 500 fs with relative amplitudes of 70% and 30% were observed.12 In B850 a somewhat shorter initial decay of ∼50 fs was observed (70%) followed by a 500 fs anisotropy decay.13 In both cases the initial values were near 0.4 and decayed to asymptotic values near 0.1, indicating randomization of the excitation in a plane. These results were modeled by assuming Fo¨rster excitation transfer from dimer to dimer in isolated rings where the site energies follow a Gaussian distribution around the absorption maximum. The average hopping times resulting from this model are on the order of 100 fs. Reasonable agreement between the model and data were obtained for a homogeneous line width of 250 cm-1 for the dimeric subunits and a 500 cm-1 fwhm distribution of dimer absorption energies. In the case of LH1, these values are in reasonable agreement with the ∼580 cm-1 fwhm of the room temperature absorption spectrum. The difference in depolarization times in this model mainly reflects the difference in ring sizes between LH1 and LH2, not a difference in hopping rates. Precise determination of the hopping rate requires knowledge of the ring size and a monodisperse sample, as well as a method with higher time resolution. A simple exciton calculation was performed for B850 by assuming coupling strengths of 230 and 110 cm-1 for the intradimer and interdimer electronic interactions and a 200 cm-1 Gaussian distribution of site energies.13,14 The calculated exciton level structure was in good agreement with the observed width of the k ) 0 level in low-temperature hole burning.15 Furthermore, calculation of the inverse participation ratio on the low-energy side of the band gave a value of ∼0.25, i.e., an excitation delocalized over four pigments. Similar conclusions were reached for LH1.12 These calculations only considered the effects of static disorder; i.e., phonons were ignored. It was, therefore, suggested that the combined influence of static disorder and vibrations will localize the excitation onto a dimer, thereby validating the hopping model.12,13 The manner in which delocalized states would manifest themselves in these experiments, however, is still under investigation. Another type of measurement that has given information on the extent of inhomogeneous broadening in these systems is broad-band time-resolved pump-probe.23,24 In these experiments, the LH1 antenna of Rb. sphaeroides and Rhodospirillum rubrum were nonselectively excited in the Qx spectral region. As a result of energy transfer from higher energy pigments to lower energy pigments within the band, the excitation will approach a thermal distribution. This equilibration process was measured by tracking the zero-crossing point of the transient spectrum, which, in Rb. sphaeroides LH1, shifts ∼140 cm-1 lower in energy with a 700 fs time constant.24 This observation can be explained with a model almost identical to that used for modeling the fluorescence depolarization. The magnitude and time constant of the spectral shift are consistent with a ratio of homogeneous/inhomogeneous width of ∼0.6, consistent with the values obtained from the depolarization simulations. Thus far, studies have assumed a weak-coupling Coulombic (Fo¨rster) approach to calculating the energy transfer rates. In
J. Phys. Chem. B, Vol. 101, No. 37, 1997 7351 the cases of B850 and B875 dynamics, this assumption is called into question by various observations. (1) The ∼100 fs hopping times from these models imply rather large values of electronic coupling, taking the situation in these complexes outside the realm of applicability of the very weak coupling model considered by Fo¨rster.10 (2) As mentioned already, the highresolution crystal structures do not reveal distinct segregation of the Bchl into monomers, dimers, etc. Instead, the excitations are thought to be localized as a result of energetic and dynamical considerations. A proper description of energy migration in these complexes should make a natural transition between exciton and hopping models as a function of electronic coupling, static disorder, and dynamical factors (i.e., spectral density of fluctuations). (3) For both LH1 and LH2, coherent vibrational motion has been observed on time scales that far outlast the putative hopping times. Fo¨rster theory assumes vibrationally dephased, relaxed chromophores. In light of these observations, a Fo¨rster approach may not apply to energy transfer in B850 and B875. A full description of excitation transfer in these systems must include coherence transfer. The development of Redfield theory for coherent condensed phase reactive dynamics is being pursued by Jean.27-29 Evidently, a complex mechanism of energy transfer applies to these systems. The roughly 500-fold ratio between the estimated hopping times (∼ 80 fs) in LH1 and the overall time scale for trapping at the reaction center in LH1-only mutants of 50 ps invokes the notion of a “storage ring” though, of course, we do not mean to imply a unidirectional energy flow in the ring.30,31 For such a storage ring to function efficiently, weak coupling between the electronic and nuclear degrees of freedom seems necessary to avoid excessive dissipation of the excitation energy. In a system of noninteracting chromophores, the gap between the steady state absorption and emission spectra (the Stokes shift) gives a direct estimate of the overall strength of coupling of the electronic transition with the nuclear degrees of freedom (i.e., the total reorganization from both solute and solvent). However, in an inhomogeneous system with energy transfer, the progressive red shift of the emission as the excitation “equilibrates” over the distribution makes it difficult to obtain a similar estimate of the reorganization energy. The three-pulse echo peak shift (3PEPS) measurements described in this work show that the overall reorganization is small in both B850 and B875 compared to chromophores in solution or polymer glasses.38,37 2. The 3PEPS Experiment. In this section we provide a brief description of the three-pulse echo peak shift measurement in order to elucidate the likely molecular contributions to our signals. Detailed descriptions of the experiment are given elsewhere.32 As Figure 1 shows, there are three time periods of concern here. The first time period, τ, in which the system is in an electronic superposition state, is scanned. During the second time period, T, the system is in a diagonal (population) state. This is the key point that gives the measurement its large dynamic range, extending from a few femtoseconds to hundreds of picoseconds. During the last time period, the system is in a coherence until the echo emerges. The detector integrates over the final time interval by recording the echo intensity along the phase matching direction. The observable of interest is the location of the echo maximum (obtained by scanning τ) with respect to zero delay for different fixed values of T, the population period (i.e., the “peak shift”). As discussed in ref 35, when the inhomogeneous broadening is neither extremely large nor extremely small, the shift of the echo maximum from zero, τ*(T), at a particular value of T reflects the extent to which rephasing is still possible. In other words, the peak shift reflects the degree of retained memory of the initial transition frequency.
7352 J. Phys. Chem. B, Vol. 101, No. 37, 1997
Jimenez et al. and
τ*(Tf∞) )
Figure 1. Pulse sequence and phase matching diagram for the threepulse echo. The three pulses, incident on the sample with wave vectors k1, k2, and k3, generate two mirror-image echo signals in the phase matched directions k1 - k2 + k3 and -k1 + k2 + k3, as indicated in the diagram by the open circles. The experimentally controllable centerto-center temporal delays between k1 and k2 and between k2 and k3 are τ and T, respectively. These two delays are sometimes referred to as the coherence and population times. For the purposes of calculation, t1, t2, and t3 denote field-matter interaction points, while t1′, t2′, and t3′ refer to delays between interaction points (see eq 9). P(3)(t′) is the thirdorder polarization generated by the three interactions at t1′, t2′, and t3.
In a system without energy transfer, both dynamical and static contributions to the line width are reflected in the form of τ*(T) vs T. Static inhomogeneity means that some rephasing will be possible even at long times; only when the memory of the transition frequency is completely lost will τ*(T) ) 0. In general, the memory of the electronic transition frequency can be written as a time correlation function. For the case of noninteracting molecules in solvents, the peak shift has been shown to reveal time scales in the time correlation function of the electronic transition frequency, M(t):35
M(t) )
〈∆ω(0) ∆ω(t)〉 〈∆ω2〉
(1)
where ∆ω(t) is the fluctuating part of the electronic transition frequency for each chromophore, referenced to its central frequency,
ωi(t) ) ∆ω(t) + 〈ω〉 + i
(2)
where i is the static offset (taken from the inhomogeneous distribution) and 〈ω〉 is the average electronic transition frequency.43 For a typical dye molecule in solvent at room temperature, M(t) is a sum of Gaussian (inertial), exponential (diffusive), and oscillatory (inter- and intramolecular vibrational) contributions.43 M(t) does not make a distinction between modes of the protein environment and vibrational modes of the chromophores. To extract quantitative values of the parameters from peak shift data requires extensive numerical modeling.32,33 However, the physical content of the experiment can be appreciated via two approximate expressions valid only (a) at high temperatures, (b) for times longer than the bath correlation time, and (c) for intermediate inhomogeneous broadening (∆in< Γ, see ref 35 for details).
τ*(T) )
xΓ + ∆in2 + f(T) (Γ S(T) + ∆in2) xπ[Γ(Γ + 2∆in2 + f(T)) + ∆in2f(T)]
(3)
∆in2xΓ + ∆in2 + (λ/p)2
xπ[Γ(Γ + 2∆in2 + (λ/p)2) + ∆in2(λ/p)2]
(4)
where Γ ) 2λ/p2β (β ) 1/kT), λ is the reorganization energy (2λ is the static Stokes shift), ∆in is the fwhm of the inhomogeneous distribution, and S(T) is the Stokes shift function, which at high temperature is equal to M(t). Finally, f(T) ) (λ/p)2[1 - M(T)/M(0)]2. Thus, the sole dynamical quantity appearing in the peak shift is S(t) or M(t), while at long times the peak shift is determined by the ratio of fast broadening (λ) and the static disorder (∆in). If ∆in is zero, τ*(Tf∞) ) 0, as can be readily seen from eq 4. We now consider how expressions 3 and 4 (or the exact formulas used for data fitting) would be modified in the presence of energy transfer. The rapid processes leading to loss of memory of the transition frequency (i.e., intramolecular vibrational dynamics and inertial solvation) will still be present in the signal, as in isolated chromophores. The key difference lies in the role of the inhomogeneous broadening. This can be seen by examining eq 2. The reason that static inhomogeneous broadening allows rephasing is that individual chromophores retain memory of their transition frequencies. We are able to label each chromophore with a time-independent offset, i, because all the optically excited chromophores remain excited. Energy transfer destroys the memory, probably on the time scale of one or a few hops, because the label i now tracks the electronically excited chromophore. Therefore, when energy transfer occurs, memory of the inhomogeneous distribution is not retained at all times: i becomes i(t). We, therefore, suggest that energy transfer can be incorporated by considering ∆in to be time dependent; its ability to allow rephasing is progressively destroyed. The simplest assumption is that this process is exponential, and this is equivalent to adding an exponential term to M(t), whose amplitude is related to the width of the distribution. The energy transfer plays a role somewhat analogous to motional narrowing. In this case it is clear that the τ*(Tf∞) values from these systems with extensive energy transfer cannot be used to obtain the inhomogeneous width directly. The 3PEPS experiment gives a direct measure of the homogeneous width, which in turn enables determination of the inhomogeneous width from the width of the absorption spectrum. In the absence of a more formal theory of the 3PEPS experiment in the presence of energy transfer, we take the approach outlined above for the analysis of our data. II. Experimental Section These experiments were performed on standard, detergentisolated LH1 and LH2 complexes of Rhodobacter sphaeroides, solubilized in N-octyl-β-D-glucopyranoside. The samples were chilled with an ice bath and flowed through a 200 µm quartz cell with a peristaltic pump. The laser used for these experiments is a self-mode-locked Ti:sapphire laser pumped by ∼6 W from an Ar+ laser. The mode-locked output could be tuned from ∼840 to 900 nm with a razor blade inserted in the dispersed end of the cavity. The laser was cavity dumped at 250 kHz repetition rate so as to avoid the buildup of triplet Bchl in the complexes. The pulse widths were 30-35 fs at these wavelengths. The total pulse energy in each beam used for the echo measurements was in the range from 800 pJ to 1 nJ, as measured with a power meter at the sample position. The peak shift measurement was performed as previously described, with a standard three-beam optical arrangement.32
LH1 and LH2 Complexes of Rhodobacter sphaeroides
Figure 2. Three-pulse echo decays for LH1 at population delays T ) 0 and T ) 1000 fs, shown for the two phase matched directions. The peak shift is half the time difference between the peaks of the two echo decays at each T.
Simultaneous resolution of the two phase-matched echo signals (k1 - k2 + k3 and -k1 + k2 + k3) allows precise determination of the peak shift (error bars ( 0.3 fs). In order to minimize scatter, two beams were chopped, and lock-in detection of the diode signals at the difference frequency was employed. The beams were focused into the sample by a 20 cm achromatic lens to avoid large crossing angles between beams which may result in an artificial residual peak shift.37,38 It was verified that the peak shift for IR144 in DMSO was 25 ps. This observation is important for the interpretation of residual peak shift values. Normally, the three-pulse echo peak shift measurement is performed with all polarizations parallel to each other. In this study, we have an interesting situation where the excitation moves among chromophores whose transition moments are not parallel. One might anticipate polarization effects on the echo signals. The typical coherence period (see Figure 1) is of a duration comparable or smaller than the observed depolarization times. However, the population periods are often of much longer duration. We set the polarizations of the first two pulses parallel to each other and varied the polarization of the third pulse (k3) from parallel to magic angle. We found only a small effect on the peak shift as a result of the polarization changes. The magnitude of the peak shift decreased by ∼1 fs or less at all times when rotating the polarization from parallel to magic angle, and the time scales of the decays were not affected. III. Results and Discussion The LH1 and LH2 transient absorption, transient grating, and three-pulse echo signals were measured in this study. Representative LH1 three pulse echo decays for population times of T ) 0 fs and T ) 1000 fs are shown in Figure 2. The LH1 and LH2 peak shifts, each measured with the polarization of k3 at magic angle, are shown in Figure 3. As a starting point for discussion, we note the interesting features of these peak shift decays. (1) The initial values of the peak shifts (∼24 fs) are in both cases significantly larger than typically observed for dyes.32,38 (2) The peak shift decays are 90% complete in 50 fs (for reasons to be discussed below, we find discrepancies at early times). The fit at early times is improved by decreasing the time scale of the Gaussian component from 90 fs down to 60 fs, but no improvement occurs for shorter time constants. The LH1 data and simulated peak shift are shown in Figure 5, along with M(t). The simulated and observed spectra are shown in Figure 6. The values of the parameters for M(t) are contained in Table 1.
LH1 and LH2 Complexes of Rhodobacter sphaeroides
J. Phys. Chem. B, Vol. 101, No. 37, 1997 7355
Figure 5. (top) LH1 experimental peak shift (symbols) and calculated (line) peak shift using 35 fs pulses. (bottom) Model M(t) used for the calculation.
Figure 7. (top) LH2 experimental peak shift (symbols) and calculated peak shift using 35 fs pulses. (bottom) Model M(t) used for the calculation.
Figure 6. Measured (symbols) and simulated (line) absorption spectra for LH1. The simulated spectrum was calculated using the procedure described in the text, with ωeg ) 11 320 cm-1.
Figure 8. Measured (symbols) and simulated (line) absorption spectra for LH2. The simulated spectrum was calculated using the procedure described in the text, with ωeg ) 11 710 cm-1.
TABLE 1: Parameters for LH1 Model M(t); See Eq 4 for Definitions
TABLE 2: Parameters for LH2 Model M(t); See Eq 4 for Definitions
Gaussian exponential exponential2 oscillation 1 oscillation 2 oscillation 3 oscillation 4
〈∆ω2〉1/2 (cm-1)
τ (fs)
100 160 85 45 45 170 120
60 90 12500
ν (cm-1) τdamp (fs) φ (rad)
110 190 560 750
700 400 70 50
-2.1 0 -0.2 0
A similar procedure was performed in fitting the LH2 peak shift (data, fit, and M(t) are shown in Figure 7; the parameters are collected in Table 2). The slow phase of the decay is of smaller amplitude and longer time constant than in LH1. Also, the coupling strengths of the vibrations were decreased in order to account for their smaller amplitudes in the LH2 peak shift and transient grating signal. Furthermore, the frequencies of two excited state fundamental vibrations (750 and 920 cm-1) determined by low-temperature hole-burning experiments were included in M(t).16 The coupling strengths were calculated from
Gaussian exponential exponential2 oscillation 1 oscillation 2 oscillation 3 oscillation 4
〈∆ω2〉1/2 (cm-1)
τ (fs)
80 170 65 30 30 190 190
40 130 15000
ν (cm-1) τdamp (fs) φ (rad)
110 190 750 920
700 400 50 50
-1.5 0 0 0
the Franck-Condon factors (for the 0-1 transition) given in reference 16. The fit to the LH2 absorption spectrum (Figure 8) is rather good. Attempts to ascertain the quality of the fit are complicated by the B800 band on the high-energy side of the B850 peak, making it difficult to determine the shape of the B850 spectrum in this region in the absence of B800. However, mutant LH2 complexes are available which are thought to lack the binding site for B800.26 The absorption spectra of these complexes are very similar to our simulated spectrum. Note that both our simulated line shapes and low-
7356 J. Phys. Chem. B, Vol. 101, No. 37, 1997 temperature hole burning indicate the presence of B850 absorption in the B800 region. This absorption will facilitate B800 to B850 energy transfer by providing spectral overlap. Furthermore, this absorption will complicate the analysis of transient absorption measurements which assume that only B800 absorbs at these wavelengths. Our model M(t) functions give estimated displacements between absorption and emission spectra of 212 and 160 cm-1 for LH1 and LH2, respectively. In order to determine the Stokes shift in the absence of energy transfer, one must remove the exponential components from consideration. In doing so, we obtain values of 70 cm-1 for LH1 and 40 cm-1 for LH2. Within a strongly coupled dimer model of B850 and B875, one would equate these Stokes shift values with those of the dimeric subunits comprising each ring. The observed displacement between absorption and emission maxima of the B820 subunit from Rs. Rubrum is 66 cm-1 at 77 K.11 Recent pump-probe experiments by Kumble et al. on B820 have led to an estimate of the value of the Stokes shift of ∼80 cm-1.40 For comparison, the Stokes shift of Bchl dissolved in cyclohexanol has been measured to be 260 cm-1.36 The remaining portion of the shift due to the exponential components can be loosely associated with the spectral equilibration due to energy transfer. The magnitude of these shifts are 140 cm-1 for LH1 and 120 cm-1 for LH2. The value for LH1 is in good agreement with that observed for spectral equilibration in broad-band pump-probe measurements and with the value of spectral shift expected from room temperature equilibration of a 500 cm-1 fwhm distribution of pigment energies.24 Another contribution to the Stokes shift which may be important is that arising from relaxation among the exciton levels of the aggregate. Recently, Meier et al. have estimated this shift to be 139 cm-1.4 Later, we will consider the consequences of the magnitude of the Stokes shift on the electronic level structure of the Bchl aggregates. The fast (40-60 fs) Gaussian component in M(t) comes from rapid fluctuations of the protein environment, a process that we choose to call homogeneous broadening. Of course, there are many points of view on the meaning of “homogeneous”, but we state here a definition that suffices for a clear discussion in the present inquiry. In this view, the absorption spectrum is composed of intramolecular vibrational lines, each of which is broadened by the dynamical processes of the medium (in this case, the protein). The fluctuations of the protein are of sufficiently low frequency that they are thermally active, thus satisfying the fluctuation-dissipation theorem; under these conditions the homogeneous line width is related to the Stokes shift as stated in eq 5. To make this definition clear, we restate the point: the homogeneous line width is only that caused by the dynamics (vibrations) of the protein or solvent environment. Thus, the coupling strength of the Gaussian component in M(t), multiplied by x(8 ln 2), is the fwhm of the homogeneous line width. Implicit in our definition is the assumption that we can somehow distinguish between the vibrational modes of the chromophore and those of the environment. This distinction cannot be made by an examination of M(t), because all modes are added in the same manner. Therefore, another possible definition for the fwhm of the homogeneous line width, which includes all thermally active modes, is the sum of the coupling strengths for all modes with frequencies 200 cm-1 (see Tables 1 and 2) to calculate the thermal fluctuation amplitude. We find Γ ) 447 cm-1 for LH1 and 239 cm-1 for LH2. These values are comparable to the magnitude of the electronic coupling in B850, as estimated by point-monopole calculations (J ∼ 250 cm-1).13,20 Next, we utilize Leegwater’s approach as a means of estimating the combined effect of fluctuations and disorder.22 In that work, the effect of vibrations is represented by Γ. Furthermore, our results show that the standard deviation of the inhomogeneous line width (∆) is 530 cm-1/x 8 ln 2, or 226 cm-1. Summarizing our results, we have Γ/J ∼ 2, and ∆/J ∼ 1, which implies an excitation localized on 3-4 pigments (see Figure 6 of ref 22). Similar considerations hold for B850. An important consequence of our analysis is the finding that the vibrational coupling strengths outweigh the electronic coupling in LH1 and LH2; i.e., the electronic states are dynamically localized. Kumble et al. take a similar view of the issue of dynamic localization.40 In their study, the time scale of the B820 Stokes shift is evident in their experiment as a 20 fs movement of the nuclear wavepacket, which causes both a shift in their frequencyresolved transient absorption signals and a fast decay component in the anisotropy. These vibrational dynamics contribute to the initial decay of the peak shift; both measurements reveal the time scale of the same process.34 This time scale of the Stokes shift is much shorter than the estimated hopping times in B850 and B875 (see below). As a consequence, even if the electronic state initially prepared by a short optical pulse is delocalized over many Bchl, it will rapidly localize onto 3-4 pigments, so that a weak-coupling approach should be appropriate at times J50 fs. Kumble and co-workers note that the 80 cm-1 B820 Stokes shift implies a 300 cm-1 homogeneous line width and
7358 J. Phys. Chem. B, Vol. 101, No. 37, 1997 point out that the excited states of LH1 must be localized unless the value of the electronic coupling is significantly larger than 300 cm-1. As indicated by this discussion, recent results on the subject of localization in antenna systems provide more impetus for the view that excitation transfer within B850 and B875 can be modeled well with dimeric excitations. 4. Relation to Previous Work; Fluorescence Depolarization. When using the hopping model, the ∼100 fs decay process in M(t) may be interpreted as follows: In the absence of siteenergy correlations between neighboring pigments, “hopping” on an inhomogeneously broadened ring will cause a decay of M(t) on time scales related to the hopping time. More specifically, M(t) will track the probability that the excitation remains on the initially excited pigment (and electronic state). In the case of a homogeneous system, the rate matrix for a ring with only nearest-neighbor hopping can be diagonalized analytically, revealing multiexponential behavior.49 When approximating this multiexponential decay with a biexponential function, the major component is approximately half the hopping time. Within this interpretation of M(t), the 90 fs decay of M(t) in LH1, for example, indicates a hopping time of ∼180 fs. For an inhomogeneous ring, one has to resort to Monte Carlo calculations; however, the fast lifetimes will fall between τhop/2 and τhop. For rings larger than four sites, these conclusions are not sensitive to the ring size; the ring size mainly affects the amplitudes of the slower components. Numerical simulations and more detailed discussion of these results will be presented in a separate paper.50 We can also estimate the homogeneous transfer time using Fo¨rster theory. The values for the B850 and B875 Stokes shifts and homogeneous line shapes are taken from the present work. For excitation transfer on the time scale of the vibrational dynamics, the use of the homogeneous line shape for the overlap integral is problematic;10 use of the strong-coupling or intermediate-coupling cases is more appropriate. However, we will use the weak-coupling formula simply as an estimate of the hopping time scales. The integrated transition dipole strength of the Bchl dimer is taken to be 77 D2.12 LH1 is modeled as a 96 Å diameter ring of 16 dimers whose transition dipoles are tangential to the ring.6 The B850 pigments are modeled as a 52 Å diameter ring of nine dimers whose transition dipoles are tilted 22° with respect to the ring. This arrangement is consistent with the 2.5 Å crystal structure of LH2 from Rps. acidophila.4 The homogeneous hopping times, as calculated from Fo¨rster theory, are 75 and 80 fs for B875 and B850, respectively. These hopping times give depolarization times of 128 and 50 fs for B875 and B850, respectively. These depolarization time scales are in reasonable agreement with those found in refs 12 and 13. The estimate of the LH1 depolarization time is slightly longer than the dominant value found in ref 12. Part of this discrepancy may result from the assumption of a 16-dimer ring for LH1; smaller rings would lead to better agreement. Both time-resolved annihilation experiments12 and two-dimensional diffraction studies6 have shown evidence for smaller rings. These considerations indicate that the dimer-to-dimer hopping model, utilized with the parameters determined by this echo study, is consistent with previous work on the B850 and B875 bands. In particular, the hopping times and inhomogeneous bandwidths resulting from our model are consistent with earlier studies.12,13,24 The most likely explanation for the exponential decay time scales in M(t), therefore, seems to be the decay of population from the initially excited chromophores. It is striking that the M(t) functions for B850 and B875, in contrast with those for chromophores in solvents and glasses, decay so rapidly.32,37,38 We suggest that longer time scale dynamics are inaccessible to the B850 and B875 3PEPS measurements
Jimenez et al. because the ability for the system to rephase is lost on the time scale of the fast hopping process. We will look for the slower protein contributions to M(t) in these antenna complexes by performing these same experiments on the isolated B820 dimer from the LH1 antenna. IV. Concluding Remarks We summarize our findings on the parameters underlying B875 and B850 dynamics, based on the model M(t) functions that reproduce the peak shifts and absorption spectra: For LH1 we find a homogeneous line width of 235 cm-1, an inhomogeneous line width of 530 cm-1, a Stokes shift of 212 cm-1, and an electronic origin (0-0 transition) at 11 320 cm-1. For LH2 we find a homogeneous line width of 188 cm-1, an inhomogeneous line width of 390 cm-1, a Stokes shift of 165 cm-1, and an electronic origin at 11 710 cm-1. In both cases, we do not include the presence of multiple exciton states; the inhomogeneous line widths should thus be considered upper limits. For both LH1 and LH2 we find high-frequency vibrational modes to be of importance for simulating the peak shift and absorption spectra. In the case of LH2, we include 750 and 920 cm-1 vibrations observed in B850 hole-burning studies.16 For LH1, we model the influence of nonimpulsively excited vibrations by using frequencies of 560 and 750 cm-1. The thermally active vibrational modes, in combination with the Gaussian homogeneous process in M(t), serve to dynamically localize the electronic states in B850 and B875. The model M(t)s reveal that rapid fluctuations of magnitude comparable to the electronic coupling are present in B850 and B875. We followed a simplified treatment to provide an estimate for the combined effects of static disorder and thermally active vibrations and find the electronic states localized onto 3-4 Bchl.22 Finally, we show that the hopping model, when applied to LH1, is consistent with previous time-resolved experimental results. Similar results hold for LH2. The M(t) functions for B850 and B875 will be of use as input to theoretical treatments of energy transfer that utilize a spectral density to model the coupling between the electronic and nuclear degrees of freedom.51-53 Of course, a formal theoretical framework for the effect of energy transfer on the echo signal remains to be developed. Challenges for a description of the echo signal in the presence of energy transfer include the need to account for the progressive loss of rephasing due to incoherent hopping, input of the excitation conditions, evolution of the inhomogeneous distribution, and multiple excitonic states. The collective oscillator density matrix approach very recently presented by Meier et al. provides a framework for the description of these dynamics.41 At present, this approach is able to consistently explain both the temperature-dependent superradiance21 and pump-probe experiments.19 The current experimental results provide further evidence that vibrational coherence transfer is at work in LH1 and LH2, because the hopping times derived from the model are much shorter than the dephasing of the coherences for the 110 and 190 cm-1 vibrational modes. The study of coherence transfer in LH1 and LH2 is an important direction for the future. We conclude by noting that the total magnitude of the fluctuations 〈∆ω2 〉 for B850 and B875 (as judged by their Stokes shifts and homogeneous line widths) is much smaller than that observed for dyes in polar solvents or glasses in similar studies. In comparison, the electronic transitions of Bchl in light-harvesting systems seem to be more weakly coupled both to fluctuations in the protein environment and to intramolecular vibrations. This weak system-bath coupling is likely to be one of the most important factors underlying the efficiency of the energy transfer processes in LH1 and LH2 complexes.
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